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logistic-regressors/logistic-regressor.ipynb

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None <html> <head> </head>
In [1]:
from sklearn import datasets
iris = datasets.load_iris()
list(iris.keys())
Out[1]:
['data',
 'target',
 'frame',
 'target_names',
 'DESCR',
 'feature_names',
 'filename',
 'data_module']
In [2]:
iris
Out[2]:
{'data': array([[5.1, 3.5, 1.4, 0.2],
        [4.9, 3. , 1.4, 0.2],
        [4.7, 3.2, 1.3, 0.2],
        [4.6, 3.1, 1.5, 0.2],
        [5. , 3.6, 1.4, 0.2],
        [5.4, 3.9, 1.7, 0.4],
        [4.6, 3.4, 1.4, 0.3],
        [5. , 3.4, 1.5, 0.2],
        [4.4, 2.9, 1.4, 0.2],
        [4.9, 3.1, 1.5, 0.1],
        [5.4, 3.7, 1.5, 0.2],
        [4.8, 3.4, 1.6, 0.2],
        [4.8, 3. , 1.4, 0.1],
        [4.3, 3. , 1.1, 0.1],
        [5.8, 4. , 1.2, 0.2],
        [5.7, 4.4, 1.5, 0.4],
        [5.4, 3.9, 1.3, 0.4],
        [5.1, 3.5, 1.4, 0.3],
        [5.7, 3.8, 1.7, 0.3],
        [5.1, 3.8, 1.5, 0.3],
        [5.4, 3.4, 1.7, 0.2],
        [5.1, 3.7, 1.5, 0.4],
        [4.6, 3.6, 1. , 0.2],
        [5.1, 3.3, 1.7, 0.5],
        [4.8, 3.4, 1.9, 0.2],
        [5. , 3. , 1.6, 0.2],
        [5. , 3.4, 1.6, 0.4],
        [5.2, 3.5, 1.5, 0.2],
        [5.2, 3.4, 1.4, 0.2],
        [4.7, 3.2, 1.6, 0.2],
        [4.8, 3.1, 1.6, 0.2],
        [5.4, 3.4, 1.5, 0.4],
        [5.2, 4.1, 1.5, 0.1],
        [5.5, 4.2, 1.4, 0.2],
        [4.9, 3.1, 1.5, 0.2],
        [5. , 3.2, 1.2, 0.2],
        [5.5, 3.5, 1.3, 0.2],
        [4.9, 3.6, 1.4, 0.1],
        [4.4, 3. , 1.3, 0.2],
        [5.1, 3.4, 1.5, 0.2],
        [5. , 3.5, 1.3, 0.3],
        [4.5, 2.3, 1.3, 0.3],
        [4.4, 3.2, 1.3, 0.2],
        [5. , 3.5, 1.6, 0.6],
        [5.1, 3.8, 1.9, 0.4],
        [4.8, 3. , 1.4, 0.3],
        [5.1, 3.8, 1.6, 0.2],
        [4.6, 3.2, 1.4, 0.2],
        [5.3, 3.7, 1.5, 0.2],
        [5. , 3.3, 1.4, 0.2],
        [7. , 3.2, 4.7, 1.4],
        [6.4, 3.2, 4.5, 1.5],
        [6.9, 3.1, 4.9, 1.5],
        [5.5, 2.3, 4. , 1.3],
        [6.5, 2.8, 4.6, 1.5],
        [5.7, 2.8, 4.5, 1.3],
        [6.3, 3.3, 4.7, 1.6],
        [4.9, 2.4, 3.3, 1. ],
        [6.6, 2.9, 4.6, 1.3],
        [5.2, 2.7, 3.9, 1.4],
        [5. , 2. , 3.5, 1. ],
        [5.9, 3. , 4.2, 1.5],
        [6. , 2.2, 4. , 1. ],
        [6.1, 2.9, 4.7, 1.4],
        [5.6, 2.9, 3.6, 1.3],
        [6.7, 3.1, 4.4, 1.4],
        [5.6, 3. , 4.5, 1.5],
        [5.8, 2.7, 4.1, 1. ],
        [6.2, 2.2, 4.5, 1.5],
        [5.6, 2.5, 3.9, 1.1],
        [5.9, 3.2, 4.8, 1.8],
        [6.1, 2.8, 4. , 1.3],
        [6.3, 2.5, 4.9, 1.5],
        [6.1, 2.8, 4.7, 1.2],
        [6.4, 2.9, 4.3, 1.3],
        [6.6, 3. , 4.4, 1.4],
        [6.8, 2.8, 4.8, 1.4],
        [6.7, 3. , 5. , 1.7],
        [6. , 2.9, 4.5, 1.5],
        [5.7, 2.6, 3.5, 1. ],
        [5.5, 2.4, 3.8, 1.1],
        [5.5, 2.4, 3.7, 1. ],
        [5.8, 2.7, 3.9, 1.2],
        [6. , 2.7, 5.1, 1.6],
        [5.4, 3. , 4.5, 1.5],
        [6. , 3.4, 4.5, 1.6],
        [6.7, 3.1, 4.7, 1.5],
        [6.3, 2.3, 4.4, 1.3],
        [5.6, 3. , 4.1, 1.3],
        [5.5, 2.5, 4. , 1.3],
        [5.5, 2.6, 4.4, 1.2],
        [6.1, 3. , 4.6, 1.4],
        [5.8, 2.6, 4. , 1.2],
        [5. , 2.3, 3.3, 1. ],
        [5.6, 2.7, 4.2, 1.3],
        [5.7, 3. , 4.2, 1.2],
        [5.7, 2.9, 4.2, 1.3],
        [6.2, 2.9, 4.3, 1.3],
        [5.1, 2.5, 3. , 1.1],
        [5.7, 2.8, 4.1, 1.3],
        [6.3, 3.3, 6. , 2.5],
        [5.8, 2.7, 5.1, 1.9],
        [7.1, 3. , 5.9, 2.1],
        [6.3, 2.9, 5.6, 1.8],
        [6.5, 3. , 5.8, 2.2],
        [7.6, 3. , 6.6, 2.1],
        [4.9, 2.5, 4.5, 1.7],
        [7.3, 2.9, 6.3, 1.8],
        [6.7, 2.5, 5.8, 1.8],
        [7.2, 3.6, 6.1, 2.5],
        [6.5, 3.2, 5.1, 2. ],
        [6.4, 2.7, 5.3, 1.9],
        [6.8, 3. , 5.5, 2.1],
        [5.7, 2.5, 5. , 2. ],
        [5.8, 2.8, 5.1, 2.4],
        [6.4, 3.2, 5.3, 2.3],
        [6.5, 3. , 5.5, 1.8],
        [7.7, 3.8, 6.7, 2.2],
        [7.7, 2.6, 6.9, 2.3],
        [6. , 2.2, 5. , 1.5],
        [6.9, 3.2, 5.7, 2.3],
        [5.6, 2.8, 4.9, 2. ],
        [7.7, 2.8, 6.7, 2. ],
        [6.3, 2.7, 4.9, 1.8],
        [6.7, 3.3, 5.7, 2.1],
        [7.2, 3.2, 6. , 1.8],
        [6.2, 2.8, 4.8, 1.8],
        [6.1, 3. , 4.9, 1.8],
        [6.4, 2.8, 5.6, 2.1],
        [7.2, 3. , 5.8, 1.6],
        [7.4, 2.8, 6.1, 1.9],
        [7.9, 3.8, 6.4, 2. ],
        [6.4, 2.8, 5.6, 2.2],
        [6.3, 2.8, 5.1, 1.5],
        [6.1, 2.6, 5.6, 1.4],
        [7.7, 3. , 6.1, 2.3],
        [6.3, 3.4, 5.6, 2.4],
        [6.4, 3.1, 5.5, 1.8],
        [6. , 3. , 4.8, 1.8],
        [6.9, 3.1, 5.4, 2.1],
        [6.7, 3.1, 5.6, 2.4],
        [6.9, 3.1, 5.1, 2.3],
        [5.8, 2.7, 5.1, 1.9],
        [6.8, 3.2, 5.9, 2.3],
        [6.7, 3.3, 5.7, 2.5],
        [6.7, 3. , 5.2, 2.3],
        [6.3, 2.5, 5. , 1.9],
        [6.5, 3. , 5.2, 2. ],
        [6.2, 3.4, 5.4, 2.3],
        [5.9, 3. , 5.1, 1.8]]),
 'target': array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
        0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
        0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
        2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
        2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]),
 'frame': None,
 'target_names': array(['setosa', 'versicolor', 'virginica'], dtype='<U10'),
 'DESCR': '.. _iris_dataset:\n\nIris plants dataset\n--------------------\n\n**Data Set Characteristics:**\n\n    :Number of Instances: 150 (50 in each of three classes)\n    :Number of Attributes: 4 numeric, predictive attributes and the class\n    :Attribute Information:\n        - sepal length in cm\n        - sepal width in cm\n        - petal length in cm\n        - petal width in cm\n        - class:\n                - Iris-Setosa\n                - Iris-Versicolour\n                - Iris-Virginica\n                \n    :Summary Statistics:\n\n    ============== ==== ==== ======= ===== ====================\n                    Min  Max   Mean    SD   Class Correlation\n    ============== ==== ==== ======= ===== ====================\n    sepal length:   4.3  7.9   5.84   0.83    0.7826\n    sepal width:    2.0  4.4   3.05   0.43   -0.4194\n    petal length:   1.0  6.9   3.76   1.76    0.9490  (high!)\n    petal width:    0.1  2.5   1.20   0.76    0.9565  (high!)\n    ============== ==== ==== ======= ===== ====================\n\n    :Missing Attribute Values: None\n    :Class Distribution: 33.3% for each of 3 classes.\n    :Creator: R.A. Fisher\n    :Donor: Michael Marshall (MARSHALL%PLU@io.arc.nasa.gov)\n    :Date: July, 1988\n\nThe famous Iris database, first used by Sir R.A. Fisher. The dataset is taken\nfrom Fisher\'s paper. Note that it\'s the same as in R, but not as in the UCI\nMachine Learning Repository, which has two wrong data points.\n\nThis is perhaps the best known database to be found in the\npattern recognition literature.  Fisher\'s paper is a classic in the field and\nis referenced frequently to this day.  (See Duda & Hart, for example.)  The\ndata set contains 3 classes of 50 instances each, where each class refers to a\ntype of iris plant.  One class is linearly separable from the other 2; the\nlatter are NOT linearly separable from each other.\n\n.. topic:: References\n\n   - Fisher, R.A. "The use of multiple measurements in taxonomic problems"\n     Annual Eugenics, 7, Part II, 179-188 (1936); also in "Contributions to\n     Mathematical Statistics" (John Wiley, NY, 1950).\n   - Duda, R.O., & Hart, P.E. (1973) Pattern Classification and Scene Analysis.\n     (Q327.D83) John Wiley & Sons.  ISBN 0-471-22361-1.  See page 218.\n   - Dasarathy, B.V. (1980) "Nosing Around the Neighborhood: A New System\n     Structure and Classification Rule for Recognition in Partially Exposed\n     Environments".  IEEE Transactions on Pattern Analysis and Machine\n     Intelligence, Vol. PAMI-2, No. 1, 67-71.\n   - Gates, G.W. (1972) "The Reduced Nearest Neighbor Rule".  IEEE Transactions\n     on Information Theory, May 1972, 431-433.\n   - See also: 1988 MLC Proceedings, 54-64.  Cheeseman et al"s AUTOCLASS II\n     conceptual clustering system finds 3 classes in the data.\n   - Many, many more ...',
 'feature_names': ['sepal length (cm)',
  'sepal width (cm)',
  'petal length (cm)',
  'petal width (cm)'],
 'filename': 'iris.csv',
 'data_module': 'sklearn.datasets.data'}
In [3]:
print(iris.DESCR)

.. _iris_dataset:

Iris plants dataset
--------------------

**Data Set Characteristics:**

    :Number of Instances: 150 (50 in each of three classes)
    :Number of Attributes: 4 numeric, predictive attributes and the class
    :Attribute Information:
        - sepal length in cm
        - sepal width in cm
        - petal length in cm
        - petal width in cm
        - class:
                - Iris-Setosa
                - Iris-Versicolour
                - Iris-Virginica
                
    :Summary Statistics:

    ============== ==== ==== ======= ===== ====================
                    Min  Max   Mean    SD   Class Correlation
    ============== ==== ==== ======= ===== ====================
    sepal length:   4.3  7.9   5.84   0.83    0.7826
    sepal width:    2.0  4.4   3.05   0.43   -0.4194
    petal length:   1.0  6.9   3.76   1.76    0.9490  (high!)
    petal width:    0.1  2.5   1.20   0.76    0.9565  (high!)
    ============== ==== ==== ======= ===== ====================

    :Missing Attribute Values: None
    :Class Distribution: 33.3% for each of 3 classes.
    :Creator: R.A. Fisher
    :Donor: Michael Marshall (MARSHALL%PLU@io.arc.nasa.gov)
    :Date: July, 1988

The famous Iris database, first used by Sir R.A. Fisher. The dataset is taken
from Fisher's paper. Note that it's the same as in R, but not as in the UCI
Machine Learning Repository, which has two wrong data points.

This is perhaps the best known database to be found in the
pattern recognition literature.  Fisher's paper is a classic in the field and
is referenced frequently to this day.  (See Duda & Hart, for example.)  The
data set contains 3 classes of 50 instances each, where each class refers to a
type of iris plant.  One class is linearly separable from the other 2; the
latter are NOT linearly separable from each other.

.. topic:: References

   - Fisher, R.A. "The use of multiple measurements in taxonomic problems"
     Annual Eugenics, 7, Part II, 179-188 (1936); also in "Contributions to
     Mathematical Statistics" (John Wiley, NY, 1950).
   - Duda, R.O., & Hart, P.E. (1973) Pattern Classification and Scene Analysis.
     (Q327.D83) John Wiley & Sons.  ISBN 0-471-22361-1.  See page 218.
   - Dasarathy, B.V. (1980) "Nosing Around the Neighborhood: A New System
     Structure and Classification Rule for Recognition in Partially Exposed
     Environments".  IEEE Transactions on Pattern Analysis and Machine
     Intelligence, Vol. PAMI-2, No. 1, 67-71.
   - Gates, G.W. (1972) "The Reduced Nearest Neighbor Rule".  IEEE Transactions
     on Information Theory, May 1972, 431-433.
   - See also: 1988 MLC Proceedings, 54-64.  Cheeseman et al"s AUTOCLASS II
     conceptual clustering system finds 3 classes in the data.
   - Many, many more ...
In [4]:
import matplotlib.pyplot as plt
import numpy as np
In [5]:
lp=iris.data[:,0:1]
wp=iris.data[:,1:2]
plt.plot(lp,wp,'.k')
Out[5]:
[<matplotlib.lines.Line2D at 0xffff6fa07580>]
In [6]:
plt.plot(lp,'.k')
Out[6]:
[<matplotlib.lines.Line2D at 0xffff6d90d610>]
In [7]:
iris.target
Out[7]:
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2])

Logistic regressor

Decision boudaries

In [8]:
import numpy as np
import matplotlib.pyplot as plt
t=np.linspace(-100, 100, 1000)
sig=1/(1+np.exp(-t+3))
plt.plot(t,sig, '.b', label=r"$\sigma=\frac{1}{1+e^{-t}}$")
plt.legend(loc='upper left', fontsize=20)
plt.axis([-5, 5, -0.1, 1])
plt.plot([0,0], [0,1], 'r--')
Out[8]:
[<matplotlib.lines.Line2D at 0xffff6d81fd30>]

Iris-Setosa Classifier based on petal width

In [9]:
X=iris.data[:,1:2]
y=(iris.target==0).astype(int)
In [10]:
from sklearn.linear_model import LogisticRegression
mylr=LogisticRegression(solver='lbfgs', random_state=42)
mylr.fit(X,y)
Out[10]:
LogisticRegression(random_state=42)
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
LogisticRegression(random_state=42)
In [11]:
Xnew=np.linspace(2,5,70).reshape(-1,1)
yPred=mylr.predict_proba(Xnew)
plt.plot(Xnew,yPred[:,0],'.r', label='No Iris-Set')
plt.plot(Xnew,yPred[:,1],'.b', label='Iris-Set')
plt.legend()
plt.plot(X,y,'*g')
Out[11]:
[<matplotlib.lines.Line2D at 0xffff6ccfc7f0>]
In [ ]:
In [12]:
# Iris-Setosa Classifier based on petal width

X=iris.data[:,0:1]
y=(iris.target==0).astype(int)

from sklearn.linear_model import LogisticRegression
mylr=LogisticRegression(solver='lbfgs', random_state=42)
mylr.fit(X,y)

Xnew=np.linspace(2,8,70).reshape(-1,1)
yPred=mylr.predict_proba(Xnew)
#plt.plot(Xnew,yPred[:,0],'.r', label='No Iris-Set')
plt.plot(Xnew,yPred[:,1],'.b', label='Iris-Set')
plt.legend()
plt.plot(X,y,'*g')
Out[12]:
[<matplotlib.lines.Line2D at 0xffff6cc07130>]

Session 2: Softmax regression

Multiple features binary classifier (Virginica)

In [13]:
import matplotlib.pyplot as plt
pl=iris.data[:,2:3]
pw=iris.data[:,3:]
tg = iris.target
plt.plot(pl[tg==0,0],pw[tg==0,0],'.r',label='Set')
plt.plot(pl[tg==1,0],pw[tg==1,0],'.b',label='Ver')
plt.plot(pl[tg==2,0],pw[tg==2,0],'.g',label='Vir')
plt.legend()
plt.show()
In [14]:
from sklearn.linear_model import LogisticRegression
X=iris.data[:,2:]
y=(iris.target==2).astype(int)

lrvir=LogisticRegression(random_state=42, tol=1e-5, C=10, max_iter=100, solver='newton-cg')
lrvir.fit(X,y)
Out[14]:
LogisticRegression(C=10, random_state=42, solver='newton-cg', tol=1e-05)
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
LogisticRegression(C=10, random_state=42, solver='newton-cg', tol=1e-05)
In [15]:
import numpy as np
x0, x1=np.meshgrid(
    np.linspace(1,6.9,500).reshape(-1,1),
    np.linspace(0.1,2.5,200).reshape(-1,1))
Xnew=np.c_[x0.ravel(), x1.ravel()]
yProb=lrvir.predict_proba(Xnew)
In [16]:
plt.figure(figsize=(10,4))
plt.plot(X[y==0,0], X[y==0,1],'bs',label='No Virg')
plt.plot(X[y==1,0], X[y==1,1],'g^',label='Virginica')
zz=yProb[:,1].reshape(x0.shape)
contour=plt.contour(x0,x1,zz)
plt.clabel(contour, inline=1,fontsize=15)
plt.xlabel("Petal Length")
plt.ylabel("Petal Width")
plt.legend()
plt.show()
In [17]:
fig, ax =plt.subplots(subplot_kw={"projection": "3d"})
surf = ax.plot_surface(x0,x1,zz)

Multiple features and multiclass classifier (Virginica)

In [35]:
X=iris.data[:,2:]
y=iris.target
In [36]:
lrmfmc=LogisticRegression(multi_class='multinomial',
                        solver='lbfgs',
                        C=100,
                        random_state=42)
lrmfmc.fit(X,y)
x0,x1=np.meshgrid(
    np.linspace(0,8,500).reshape(-1,1),
    np.linspace(0,3.5,200).reshape(-1,1))

Xnew=np.c_[x0.ravel(), x1.ravel()]
yProba=lrmfmc.predict_proba(Xnew)
#yPred=lrmfmc.predict(Xnew)

from matplotlib.colors import ListedColormap
#customc=ListedColormap(['#fafab0', '#9898ff', '#a0faa0'])
zz=yProba[:,1].reshape(x0.shape)
#zz=yPred.reshape(x0.shape)
plt.figure(figsize=(10,4))
plt.plot(X[y==2,0], X[y==2,1],'g^',label='Virg')
plt.plot(X[y==1,0], X[y==1,1],'bs',label='Versic')
plt.plot(X[y==0,0], X[y==0,1],'yo',label='Setos')
contour=plt.contour(x0,x1,zz,cmap=plt.cm.brg)
plt.clabel(contour,inline=1,fontsize=12)
#plt.contourf(x0,x1,zz,cmap=customc)
Out[36]:
<a list of 6 text.Text objects>
In [37]:
yPred=lrmfmc.predict(Xnew)
zPred=yPred.reshape(x0.shape)
plt.contourf(x0,x1,zPred,alpha=0.4)
plt.plot(X[y==2,0], X[y==2,1],'g^',label='Virg')
plt.plot(X[y==1,0], X[y==1,1],'bs',label='Versic')
plt.plot(X[y==0,0], X[y==0,1],'yo',label='Setos')
plt.show()
In [39]:
fig, ax =plt.subplots(subplot_kw={"projection": "3d"})
surf = ax.plot_surface(x0,x1,zz,cmap='jet')

Homework: Sepal multi-features and multi-class

In [20]:
X=iris.data[:,0:2]
y=iris.target
mlr3=LogisticRegression(
    multi_class='multinomial',
    solver='lbfgs',
    C=10,
    random_state=42)
mlr3.fit(X,y)
Out[20]:
LogisticRegression(C=10, multi_class='multinomial', random_state=42)
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LogisticRegression(C=10, multi_class='multinomial', random_state=42)
In [30]:
import numpy as np
x0,x1=np.meshgrid(
    np.linspace(0,7,500).reshape(-1,1),
    np.linspace(0,3,200).reshape(-1,1))
In [31]:
Xnew=np.c_[x0.ravel(), x1.ravel()]
yPred=mlr3.predict_proba(Xnew)
yProba=mlr3.predict(Xnew)
zz=yProba.reshape(x0.shape)
zz1=yPred[:,1].reshape(x0.shape)
In [32]:
from matplotlib.colors import ListedColormap
plt.plot(X[y==2,0], X[y==2,1],'y*', label='Virg')
plt.plot(X[y==1,0], X[y==1,1],'g^', label='Vers')
plt.plot(X[y==0,0], X[y==0,1],'bs', label='Set')
contour=plt.contour(x0,x1,zz1, cmap=plt.cm.brg)
ccmap=ListedColormap(['#fafab0', '#9898ff', '#a0faa0'])
plt.contourf(x0,x1,zz,cmap=ccmap)
plt.clabel(contour, inline=1,fontsize=12)
plt.xlabel('Sepal Width', fontsize=12)
plt.ylabel('Sepal Length', fontsize=12)
plt.axis([4,8.5,1,5])
plt.legend()
Out[32]:
<matplotlib.legend.Legend at 0xffff6a23b700>
In [33]:
fig, ax =plt.subplots(subplot_kw={"projection": "3d"})
surf = ax.plot_surface(x0,x1,zz1,cmap='jet')
</html>